Is there a way to understand neural networks without using the concept of brain?
tl;dr I always like to think of Neural Networks as a generalization of logistic regression. 

I too don't like that, traditionally, when introducing Neural Networks, books start with biological neurons and synapses, etc. I think its more beneficial to start from statistics and linear regression, then logistic regression and then neural networks.

A perceptron is essentially a simple binary logistic regressor (if you threshold the output). If you have many perceptrons that share the same input (i.e. a layer in a neural network), you can think of it as a multi-class logistic regressor. Now, by stacking one such layer after an other, you create a Multi-Layer Perceptron (MLP), which is a Neural Network with two layers. There is equivalent to two multi-class logistic regressors stacked one after the other. One notable thing that changes is the training technique here, i.e. backpropagation (because you don't have direct access to the targets from the hidden layer). Another thing that can change is the activation function (it's not always sigmoid in Neural Networks)

Introduce sparse connectivity and weight sharing and you get a Convolutional Neural Network. Add a connection from a layer to its self (for the next timestep) and you get a Recurrent Neural Network. Likewise, you can reproduce any Neural Network through this reasoning.

I know this is an over-simplified way of presenting them, but I think you get the point.